If $x = \frac34$ and $y = \frac43$ , find the value of $\frac12x^6y^7$.
Solution: We have  \[\frac{1}{2} x^6 y^7 = \frac{1}{2}\left(\frac{3}{4}\right)^6\left(\frac43\right)^7 = \frac{1}{2}\cdot \frac{3^6}{4^6} \cdot \frac{4^7}{3^7}
=\frac{1}{2} \cdot\frac{3^6}{3^7} \cdot \frac{4^7}{4^6} = \frac{1}{2}\cdot \frac{1}{3} \cdot 4 = \boxed{\frac{2}{3}}.\]

We also could have tackled this problem quickly by noticing that if $x=\frac34$ and $y=\frac43$, then $xy=1$, so $\frac{1}{2}x^6y^7 = \frac{1}{2} (xy)^6y=\frac{1}{2}\cdot 1^6y = \frac{1}{2}y = \frac{2}{3}$.